is a functorial homomorphism of rings ([1]}, 15.3.10).
Let \(X\)  be an \(\mathbb {F}_p\) -scheme. Let \(X_\mathrm {\acute{e}t}\)  denote the small étale site of \(X\) . For each \(n \ge 0\) , the assignment \(U \mapsto W_n(\mathcal {O}_U)\)  defines a sheaf of rings on \(X_\mathrm {\acute{e}t}\) , which we denote by \(\tilde{W}_n\) . We define \(\tilde{W} = \varprojlim _n \tilde{W}_n\) . We equip \(\tilde{W}\)  with the \(\mathbb {Z}_p\) -module endomorphism
\(\wp = F-\mathrm {id},\) 
